Found problems: 288
2022 Oral Moscow Geometry Olympiad, 5
Given a circle and a straight line $AB$ passing through its center (points $A$ and $B$ are fixed, $A$ is outside the circle, and $B$ is inside). Find the locus of the intersection of lines $AX$ and $BY$, where $XY$ is an arbitrary diameter of the circle.
(A. Akopyan, A. Zaslavsky)
1973 Spain Mathematical Olympiad, 6
An equilateral triangle of altitude $1$ is considered. For every point $P$ on the interior of the triangle, denote by $x, y , z$ the distances from the point $P$ to the sides of the triangle.
a) Prove that for every point $P$ inside the triangle it is true that $x + y + z = 1$.
b) For which points of the triangle does it hold that the distance to one side is greater than the sum of the distances to the other two?
c) We have a bar of length $1$ and we break it into three pieces. find the probability that with these pieces a triangle can be formed.
1986 Traian Lălescu, 2.2
Let be a line $ d: 3x+4y-5=0 $ on a Cartesian plane. We mark with $ \mathcal{L} $ de locus of the planar points $ P $ such that the distance from $ P $ to $ d $ is double the distance from $ P $ to the origin. Let be $ B_{\lambda } ,C_{\lambda }\in\mathcal{L} $ such that $ C_{\lambda } -B_{\lambda } +\lambda =0. $ Find the locus of the middlepoints of the segments $ B_{\lambda }C_{\lambda }, $ if $ \lambda\in\mathbb{R} $ is variable.
1954 Czech and Slovak Olympiad III A, 4
Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$.
(a) Let $M=B'Y\cap D'X$. Find the locus of all $M$s.
(b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.
1992 IMO, 1
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
V Soros Olympiad 1998 - 99 (Russia), 10.5
An isosceles triangle $ABC$ ($AB = BC$) is given on the plane. Find the locus of points $M$ of the plane such that $ABCM$ is a convex quadrilateral and $\angle MAC + \angle CMB = 90^o$.
1952 Moscow Mathematical Olympiad, 224-
You are given a segment $AB$. Find the locus of the vertices $C$ of acute-angled triangles $ABC$.
1973 IMO Shortlist, 5
A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles.
1988 Greece National Olympiad, 2
Given regular $1987$ -gon on plane with vertices $A_1, A_2,..., A_{1987}$. Find locus of points M of the plane sych that $$\left|\overrightarrow{MA_1}+\overrightarrow{MA_2}+...+\overrightarrow{MA_{1987}}\right| \le 1987$$.
1966 IMO Longlists, 55
Given the vertex $A$ and the centroid $M$ of a triangle $ABC$, find the locus of vertices $B$ such that all the angles of the triangle lie in the interval $[40^\circ, 70^\circ].$
1941 Putnam, A3
A circle of radius $a$ rolls in the plane along the $x$-axis. Show that the envelope of a diameter is a cycloid, coinciding with the cycloid traced out by a point on the circumference of a circle of diameter $a$, likewise rolling in the plane along the $x$-axis.
1978 Bulgaria National Olympiad, Problem 2
$k_1$ denotes one of the arcs formed by intersection of the circumference $k$ and the chord $AB$. $C$ is the middle point of $k_1$. On the half line (ray) $PC$ is drawn the segment $PM$. Find the locus formed from the point $M$ when $P$ is moving on $k_1$.
[i]G. Ganchev[/i]
1969 IMO Shortlist, 1
$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$
2016 Sharygin Geometry Olympiad, 3
Assume that the two triangles $ABC$ and $A'B'C'$ have the common incircle and the common circumcircle. Let a point $P$ lie inside both the triangles. Prove that the sum of the distances from $P$ to the sidelines of triangle $ABC$ is equal to the sum of distances from $P$ to the sidelines of triangle $A'B'C'$.
2005 Sharygin Geometry Olympiad, 9.3
Given a circle and points $A, B$ on it. Draw the set of midpoints of the segments, one of the ends of which lies on one of the arcs $AB$, and the other on the second.
2019 Sharygin Geometry Olympiad, 7
Let $P$ be an arbitrary point on side $BC$ of triangle $ABC$. Let $K$ be the incenter of triangle $PAB$. Let the incircle of triangle $PAC$ touch $BC$ at $F$. Point $G$ on $CK$ is such that $FG // PK$. Find the locus of $G$.
2010 Sharygin Geometry Olympiad, 6
An arbitrary line passing through vertex $B$ of triangle $ABC$ meets side $AC$ at point $K$ and the circumcircle in point $M$. Find the locus of circumcenters of triangles $AMK$.
2017 Ukrainian Geometry Olympiad, 3
Circles ${w}_{1},{w}_{2}$ intersect at points ${{A}_{1}} $ and ${{A}_{2}} $. Let $B$ be an arbitrary point on the circle ${{w}_{1}}$, and line $B{{A}_{2}}$ intersects circle ${{w}_{2}}$ at point $C$. Let $H$ be the orthocenter of $\Delta B{{A}_{1}}C$. Prove that for arbitrary choice of point $B$, the point $H$ lies on a certain fixed circle.
2006 Hanoi Open Mathematics Competitions, 6
The figure $ABCDEF$ is a regular hexagon. Find all points $M$ belonging to the hexagon such that
Area of triangle $MAC =$ Area of triangle $MCD$.
1979 Bundeswettbewerb Mathematik, 2
A circle $k$ with center $M$ and radius $r$ is given. Find the locus of the incenters of all obtuse-angled triangles inscribed in $k$.
2011 Cono Sur Olympiad, 3
Let $ABC$ be an equilateral triangle. Let $P$ be a point inside of it such that the square root of the distance of $P$ to one of the sides is equal to the sum of the square roots of the distances of $P$ to the other two sides. Find the geometric place of $P$.
2008 Switzerland - Final Round, 5
Let $ABCD$ be a square with side length $1$.
Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.
1998 Chile National Olympiad, 2
Given a semicircle of diameter $ AB $, with $ AB = 2r $, be $ CD $ a variable string, but of fixed length $ c $. Let $ E $ be the intersection point of lines $ AC $ and $ BD $, and let $ F $ be the intersection point of lines $ AD $ and $ BC $.
a) Prove that the lines $ EF $ and $ AB $ are perpendicular.
b) Determine the locus of the point $ E $.
c) Prove that $ EF $ has a constant measure, and determine it based on $ c $ and $ r $.
1969 IMO Shortlist, 12
$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.
VMEO III 2006, 11.2
Let $ABCD$ be an isosceles trapezoid, with a large base $CD$ and a small base $AB$. Let $M$ be any point on side $AB$ and $(d)$ be the line through $M$ and perpendicular to $AB$. Two rays $Mx$ and $My$ are said to satisfy the condition $(T)$ if they are symmetric about each other through $(d)$ and intersect the two rays $AD$ and $BC$ at $E$ and $F$ respectively. Find the locus of the midpoint of the segment $EF$ when the two rays $Mx$ and $My$ change and satisfy condition $(T)$.